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Unformatted text preview: Yea P Math 200 Calculus Midterm Exam I January 31, 2007 Name: _ 11300 Dec; (3.1:) i214  Cc' moi‘1 \ Lecture Hour: Section Hour: Guidelines for the test: 0 N0 books, notes, or calculators are allowed. a You may leave answers in symbolic form, like m, unless they simplify
further, like \/§ = 3,60 = 1, or cos(3ir/4) = —\/§/2. 0 Use the space provided. If necessary, write “see other side" and continue
working on the back of the same sheet. I Circle your ﬁnal answers when relevant. 0 Show all steps in your solutions and make your reasoning clear. Answers
with no explanation will receive no credit, even if they are correct. 0 No credit will be given for work/answers that are illegible 0 You have 50 minutes. Question Perfect Score Your Score /0 (l) (a) (i) Show that the following equation represents a Sphere.
2:1:24— 23:2 +222 —4x+4z = 5 2Cx1+vi+313Lx Has 5 zfxtZx +\ +y1' .;2"'+‘—:?.2“t"I "3—5 4 l "A" 2([ x— iii.4. x/z +(2+\\)‘)= Z
‘Z. '2 (x 431+ 714414 ‘3‘”: 3; "Evhs '15. om Gdtuocﬂrimﬁ 3" 951,4qu rel/v“)
tar(mat 'Wi Wuw “Hoax—i
(v—a31+(\J'b3L+L1'C317V1 (T) an—c,
Anus :3ng fawn—{1m ‘15 ‘5‘ 'i'Dlw V‘k—
(ii) What is the center and radius of the sphere ? U. gemkw (\,o,I‘j r: \?X (b) The point (1, 2, 3) rests on the surface of a Sphere of radius r centered
at the origin. Find r. Page I (2) (b) Suppose that a and b are vectors as shown below. Sketch the vector
(1 + (1, clearly showing its relation with respect to 5: and b '3"! 9l Suppose that 1'1 = (1, 1, 0) as shown below. Find and state the coor
dinates of two vectors ’5' and 13 such that 11' x 17 = 11' x 117 but 13% 11'}. :MV : (wm Mavm,‘u_.us+ Mav.,u1V1‘ 911‘“) Z Page 2‘ (3) (a) Match the foliowing equations (i)(iii) with the graphs (I)—(III). Y0u
need not explain your decisions. $213=cost y=sint, z=sin5t @ (ii) 3 = cos 2Ut, y = 81112015, 2 = logt (11) (III) (b) Sketch the curve with the vector equation
ﬁt) = t§+ e“ :3. Indicate with an arrow the direction in which 15 increases. ><
/ '2 I '2’
'f I: 7‘
2” L ‘ :3 q
l IIl  :é
 1 " 2 6””
/ \ 3. 3 .6'3 Page 3 (4) A particle moves along a. curve C with velocity at time t given by
"3(t) = (4cos t) 3+ 3}" — (4 sin 15) E.
Suppose that at time t = 0, the particle is at location given by the vector 13:41:. (3.) Find the vector functiori F(t) which gives the position of the particle
at any time t 2 0. Sleerl g
Steaml: +13” "(Lian/1W2
= can it}? +3+A§+Hcowfr¢ ...—'I {ﬁr‘1‘" em; crow/7: + big «(”03“ +26%
“7.. .V Reparametrize C with respect to arc length 5 measured from the
‘ point where t = 0 in the direction of increasing t. 3H): 3; lF‘Cm ldu (c) How far is this particle from its initial position when time t = 7: ? a a “were! , a (3%)(3):(3%) rLhliils'inﬂxi'Biif/«tiblcos 7r+qlic 0 <4 44 M“ 01" ”Tr/35+ fawn: W
vCﬂ=OA4EW§+OE cwm .: Sf _
) Find the tangential (gorﬁponent of the acceleration vector when t = ar. 21‘  v‘ “if + k: :3
Wi an r, C'L'I'ti‘xr'K v':1v'Hll:% [Limit/1 + 5,:  (Lia‘millé v l~43m+f+ a; e 4co5+ 12/ Page 4 v': l'LiS'lﬂ‘HC # qustgl
V‘ ; ”(ilel'l'xt +CO§+TZI> (5) Suppose the planes P, Q are given by
P : :1: + y — z = 2;
Q : 2m—y+32=1. (a) Let L denote the line of intersection of P and Q. Find a. point on L.
‘. '2“ 2:0
ﬂ.VW1:(”1I)(é)g<3'i:‘(3413,“l'QD swig:);,,z Yz2’)‘
(ire we? ZXV“ Yn‘V'
X_{_\/.i_2 _ 'I 2,>(:QX"'
2 F'3 "'3 ('0) 3=32< '
xii
Y: (b) Find a vector 11‘ which is parallel to L. (c) Find a. plane which is perpendicular to L and contains the point
(1,0,1). —‘éb><+8 +\/ +72 7=CJ '9X+\/ +71: ~
_ PageE...
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